A355581 Exponentially-odd 3-smooth numbers: number of the form 2^i * 3^j where i and j are either 0 or odd.

1, 2, 3, 6, 8, 24, 27, 32, 54, 96, 128, 216, 243, 384, 486, 512, 864, 1536, 1944, 2048, 2187, 3456, 4374, 6144, 7776, 8192, 13824, 17496, 19683, 24576, 31104, 32768, 39366, 55296, 69984, 98304, 124416, 131072, 157464, 177147, 221184, 279936, 354294, 393216, 497664

Offset

1,2

Links

Amiram Eldar, Table of n, a(n) for n = 1..10000

Formula

Sum_{n>=1} 1/a(n) = 55/24.

Example

6 is a term since 6 = 2^1 * 3^1 and the exponents of 2 and 3 are both odd: 1.
24 is a term since 24 = 2^3 * 3^1 and the exponents of 2 and 3 are both odd: 3 and 1, respectively.

Mathematica

q[n_] := Module[{e = IntegerExponent[n, {2, 3}]}, (e[[1]] == 0 || OddQ[e[[1]]]) && (e[[2]] == 0 || OddQ[e[[2]]]) && Times@@({2, 3}^e) == n]; Select[Range[500000], q]

Prog

(PARI) is(n) = {my(e2 = valuation(n, 2), e3 = valuation(n, 3)); (e2 == 0 || e2%2) && (e3 == 0 || e3%2) && n == 2^e2 * 3^e3};
(Python)
from itertools import count, takewhile
def aupto(lim):
    pows2 = list(takewhile(lambda x: x<lim, (2**i for i in count(1, 2))))
    pows3 = list(takewhile(lambda x: x<lim, (3**i for i in count(1, 2))))
    return sorted(c*d for c in [1]+pows2 for d in [1]+pows3 if c*d <= lim)
print(aupto(10**6)) # Michael S. Branicky, Jul 08 2022

Crossrefs

Intersection of A003586 and A268335.

Subsequences: A002023, A013711, A092810.

Cf. A355580.

Sequence in context: A095895 A051934 A153915 * A093705 A281645 A351853

Adjacent sequences: A355578 A355579 A355580 * A355582 A355583 A355584

Keyword

nonn,easy

Author

Amiram Eldar, Jul 08 2022

Status

approved